Table Of Content

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MATRIX-FREE LINEAR SYSTEM SOLVING AND APPLICATIONS TO SYMBOLIC COMPUTATION By Austin Lobo A Thesis Submitted to the Graduate Faculty of Rensselaer Polytechnic Institute in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY Major Subject: Computer Science Approved by the Examining Committee: Erich Kaltofen. Thesis Advisor W. Randolph Franklin. Member Mukkai S. Krishnamoorthy. Member David Musser, Member B. David Saunders. Member Rensselaer Polytechnic Institute Troy. New York December 1995 (For Graduation December 1995) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 9622917 Copyright 1995 by Lobo, Austin Anthony All rights reserved. UMI Microform 9622917 Copyright 1996, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. © Copyright 1995 by Austin Lobo All Rights Reserved ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CONTENTS LIST OF TABLES...................................................................................................... v LIST OF FIGURES................................................................................................... vi ACKNOWLEDGEMENT.............................................................................................vii ABSTRACT ..................................................................................................................viii 1. Wiedemann’s Coordinate Recurrence Algorithm.............................................. 1 1.1 Introduction.................................................................................................. 1 1.2 Background.................................................................................................. 3 1.3 Wiedemann’s co-ordinate recurrence algorithm...................................... 4 1.4 Homogeneous solution of linear systems................................................... 7 1.4.1 Preconditioning by Benes Networks.............................................. 9 1.5 Implementation................................................................................................12 1.6 Summ ary.........................................................................................................14 2. The Matrix-Free Block Wiedemann Algorithm....................................................16 2.1 Introduction......................................................................................................16 2.1.1 Notation................................................................................................18 2.2 Description of the Algorithm..........................................................................19 2.3 Probabilistic Justification ............................................................................25 2.4 Implementation and Program Structure......................................................26 2.5 Design of the Matrix Black B ox...................................................................30 2.6 Enhancements..................................................................................................32 2.7 Other Strategies for Distributed Execution ..............................................37 2.7.1 Black Box partitioned Transversely.................................................38 2.7.2 Black Box partitioned longitudinally.............................................39 2.7.3 Distributed Computation of the Linear Generator........................40 2.8 Fault Tolerance and Recovery from Abnormal Termination....................41 2.9 Experiments .....................................................................................................43 2.10 Summary and Concluding Remarks ............................................................49 m Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3. Rank Properties of Toeplitz M atrices...................................................................52 3.1 Introduction......................................................................................................52 3.2 Motivation.........................................................................................................54 3.3 The Extended Euclidean Algorithm.............................................................56 3.4 Subresultants...................................................................................................60 3.5 The Counting Lem m a...................................................................................62 3.6 Count of Non-Singular Toeplitz Matrices......................................................66 3.7 Generic Rank Profiles......................................................................................69 3.8 Conclusions......................................................................................................72 4. Factoring High-Degree Polynomials with a Matrix-Free Linear Solver . . . 74 4.1 Introduction......................................................................................................74 4.2 Berlekamp’s Algorithm...................................................................................78 4.3 The Black Box Algorithm.............................................................................82 4.3.1 Analysis.................................................................................................84 4.4 Implementation................................................................................................93 4.5 Future Work......................................................................................................98 5. Conclusions and Future Directions........................................................................100 6. Literature Cited.......................................................................................................103 iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF TABLES 2.1 Parallel cpu-time for solving systems modulo 32749 ................. 45 2.2 Parallel cpu-time for solving systems modulo 32749 ................. 47 2.3 Parallel cpu-time with varied grainsize for parallel computation 48 2.4 Cpu-time on SP2 architecture...........................................................48 4.1 Experiments with black box Berlekamp Algorithm.........................96 v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF FIGURES 1.1 Benes network for n = 8 .....................................................................10 1.2 Algorithm for stage s of a Benes exchange network.....................12 1.3 Software architecture of the Wiedemann CR Solver........................13 2.1 Outline of Block Wiedemann Algorithm........................................20 2.2 Finding a block linear generator........................................................23 2.3 Software Architecture of the black box Block Wiedemann Al gorithm ...................................................................................................29 2.4 Distribution Strategy for Sequence Generation................................30 2.5 Black Box representation for an M x N m atrix...........................31 2.6 C language definition of Black Box and associated variables for an Nr x Nc m atrix.......................................................................33 2.7 Initialization and Application of Black Box......................................34 2.8 Protocol for Passing a Black Box to a Program...............................35 2.9 Parallel execution with a distributed black box using trans verse partitioning...................................................................................38 2.10 Black Box partitioned transversely......................................................39 2.11 Distributed Black Box by longitudinal partitioning.....................39 2.12 Parallel execution with a black box partitioned longitudinally. 40 2.13 Parallel execution of the Minpoly phase...........................................41 4.1 Software architecture of the black box Berlekamp Factorizer. . 92 4.2 The polynomial black b o x .................................................................93 vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGEMENT I am very grateful to Erich Kaltofen who has been both mentor and friend to me. I thank him for steering me towards interesting problems and for encouraging me to tackle them. The long discussions we have had, together with his insight and his enthusiasm for research, have had a profound effect on my work. I thank the members of my doctoral committee, W. Randolph Franklin. M.S. Krishnamoorthy, David Musser, and David Saunders. Their suggestions and com ments have greatly improved the quality of this thesis. Many people have given me help in this work. I thank them all. In particular, I thank Bobby F. Caviness, Angel Dfaz, Markus Hitz, David Hollinger, Arjen Lenstra, Charles Norton, Nathan Schimke, Victor Shoup, and Thomas Yalente. I thank P. Narendran for encouraging me to return to graduate school. I gratefully acknowledge the financial assistance of the Department of Com puter Science at Rensselaer, and the funding provided by the National Science Foun dation. Finally I thank my wife Vinita for her counsel and for her love, which has always been vital to me. vii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.